Abstract

Interval temporal logics take time intervals, instead of time points, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments.

Short preliminary versions of parts of this paper appeared in [18] (all linear orders) and [2] (dense linear orders).

Notes

Acknowledgments

We thank the anonymous referees for their careful reading of our original journal submission and their insightful comments, which led to several improvements. The authors acknowledge the support from the Spanish fellowship program ‘Ramon y Cajal’ RYC-2011-07821 and the Spanish MEC project TIN2009-14372-C03-01 (G. Sciavicco), the project Processes and Modal Logics (Project No. 100048021) of the Icelandic Research Fund (L. Aceto, D. Della Monica, and A. Ingólfsdóttir), the project Decidability and Expressiveness for Interval Temporal Logics (Project No. 130802-051) of the Icelandic Research Fund in partnership with the European Commission Framework 7 Programme (People) under ‘Marie Curie Actions’ (D. Della Monica), and the Italian GNCS project Automata, Games, and Temporal Logics for the verification and synthesis of controllers in safety-critical systems (A. Montanari).

This implies that the local condition holds. As for the forward condition, consider three intervals [a, b], \([a',b']\), and [c, d] such that \([a,b]Z[a',b']\) and \([a,b]R_X[c,d]\) for some \(X \in \{ B, E, \overline{A}, \overline{E}, \overline{D} \}\). We need to exhibit an interval \([c',d']\) such that \([a',b'] R_X [c',d']\) and \([c,d] Z [c',d']\). We distinguish three cases.

If \(a > f(b)\) and \(a' > f(b')\), then we distinguish the following sub-cases.

If \(X = B\), then [c, d] is such that \(a = c < d < b\). By the monotonicity of f, we have that \(f(d) < f(b) < a = c\). Moreover, by the monotonicity of f, for every interval \([c',d']\), with \([a',b']R_B[c',d']\), \(f(d') < c'\) holds, and thus \([c,d] Z [c',d']\).

If \(X = \overline{A}\), then [c, d] is such that \(c < d = a\). Now, if \(c < f(d) = f(a)\), then, by the definition of f and Lemma 6, there exists a point \(c'\) such that \(c' < f(a') < a'\). Thus, the interval \([c',d']\), with \(d' = a'\), is such that \([a',b'] R_{\overline{A}} [c',d']\) and \([c,d] Z [c',d']\). If \(c = f(d) = f(a)\), then take \(c' =\)\(f(a')\)\(< a'\). The interval \([c',d']\), with \(d' = a'\), is such that \([a',b'] R_{\overline{A}} [c',d']\) and \([c,d] Z [c',d']\). If \(c > f(d) = f(a)\), then, by the density of \({\mathbb {R}}\), the definition of f, and Lemma 6, there exists a point \(c'\) such that \(f(a') < c' < a'\). The interval \([c',d']\), with \(d' = a'\), is such that \([a',b'] R_{\overline{A}} [c',d']\) and \([c,d] Z [c',d']\).

If \(X = \overline{E}\), then [c, d] is such that \(c < a < b = d\). There are three possibilities. If \(c < f(d)\), then, by the definition of f, there exists a point \(c'\) such that \(c' < f(b') < a'\). Thus, the interval \([c',d']\), with \(d' = b'\), is such that \([a',b'] R_{\overline{E}} [c',d']\) and \([c,d] Z [c',d']\). If \(c = f(d)\), then the interval \([c',d']\), with \(d' = b'\) and \(c' = f(d')\), is such that \([a',b'] R_{\overline{E}} [c',d']\) and \([c,d] Z [c',d']\). If \(c > f(d)\), then, by the density of \({\mathbb {R}}\), there exists a point \(c'\) such that \(f(b') < c' < a'\), and the interval \([c',d']\), with \(d' = b'\), is such that \([a',b'] R_{\overline{E}} [c',d']\) and \([c,d] Z [c',d']\).

If \(X = \overline{D}\), then [c, d] is such that \(c < a < b < d\). If \(c < f(d)\), then, take \(c' = f(a')\) and any \(d' > b'\). The interval \([c',d']\) is such that \([a',b'] R_{\overline{D}} [c',d']\) and \([c,d] Z [c',d']\). If \(c = f(d)\) (resp., \(c > f(d)\)), then, by the density of \({\mathbb {R}}\) and the monotonicity and the surjectivity of f, there exist two points \(c',d'\) such that \(c' < a' < b' < d'\) and \(c' = f(d')\) (resp., \(c' > f(d')\)). Thus, the interval \([c',d']\) is such that \([a',b'] R_{\overline{D}} [c',d']\) and \([c,d] Z [c',d']\).

If \(a < f(b)\) and \(a' < f(b')\), then we distinguish the following sub-cases.

If \(X = B\), then [c, d] is such that \(a = c < d < b\). Now, if \(c < f(d)\) (resp., \(c = f(d)\), \(c > f(d)\)), then, by the density of \({\mathbb {R}}\) and by the monotonicity and the surjectivity of f, there exists a point \(d'\) such that \(a' < d' < b'\) and \(a' < f(d')\) (resp., \(a' = f(d')\), \(a' > f(d')\)). Thus, the interval \([c',d']\), with \(c' = a'\), is such that \([a',b'] R_{B} [c',d']\) and \([c,d] Z [c',d']\).

If \(X = \overline{D}\), then [c, d] is such that \(c < a < b < d\). Thus, by the monotonicity of f, \(c < a < f(b) < f(d)\) holds. For every interval \([c',d']\), with \([a',b']R_{\overline{D}}[c',d']\), it holds, by the monotonicity of f, that \(c' < f(d')\), and thus \([c,d] Z [c',d']\).

If \(a = f(b)\) and \(a' = f(b')\), then we distinguish the following sub-cases.

If \(X = B\), then [c, d] is such that \(a = c < d < b\). Thus, \(f(d) < f(b) = a = c\) holds by the monotonicity of f. For every interval \([c',d']\), with \([a',b']R_{B}[c',d']\), by the monotonicity of f, we have that \(f(d') < c'\), and thus \([c,d] Z [c',d']\).

If \(X = E\), then [c, d] is such that \(a < c < b = d\). Thus, \(c > a = f(b) = f(d)\) holds. For every interval \([c',d']\), with \([a',b']R_{E}[c',d']\), we have that \(c' > f(d')\), and thus \([c,d] Z [c',d']\).

If \(X = \overline{A}\), then the same argument of the case when \(a > f(b)\) and \(a' > f(b')\) (and \(X = \overline{A}\)) applies.

If \(X = \overline{D}\), then [c, d] is such that \(c < a < b < d\). Thus, \(c < a = f(b) < f(d)\) holds by the monotonicity of f. For every interval \([c',d']\), with \([a',b']R_{\overline{D}}[c',d']\), by the monotonicity of f, we have that \(c' < f(d')\), and thus \([c,d] Z [c',d']\).

This implies that the local condition is satisfied. As for the forward condition, consider three intervals [a, b], \([a',b']\), and [c, d] such that \([a,b]Z[a',b']\) and \([a,b]R_X[c,d]\) for some \(X \in \{ O, \overline{B}, \overline{E}, \overline{O} \}\). We need to exhibit an interval \([c',d']\) such that \([a',b'] R_X [c',d']\) and \([c,d] Z [c',d']\). We distinguish three cases.

If \(-a > b\) and \(-a' > b'\), then, as a preliminary step, we show that the following facts hold: \((i)\, a < 0\) and \(a' < 0\); \((ii)\,|a| > |b|\) and \(|a'| > |b'|\). We only show the proofs for \(a < 0\) and \(|a| > |b|\) and we omit the ones for \(a' < 0\) and \(|a'| > |b|\), which are analogous. As for the former claim above, it is enough to observe that, if \(a \ge 0\), then \(a \ge 0 \ge -a > b\), which implies \(b < a\), leading to a contradiction with the fact that [a, b] is an interval (thus \(a < b\)). Notice that, as an immediate consequence, we have that \(|a| = -a\) holds. As for the latter claim above, firstly we suppose, by contradiction, that \(|a| = |b|\) holds. Then, \(-a = |a| = |b|\) holds and this implies either \(b = -a\), contradicting the hypothesis that \(-a > b\), or \(b = a\), contradicting the fact that [a, b] is an interval. Secondly, we suppose, again by contradiction, that \(|a| < |b|\) holds. Then, by the former claim, we have that \(0 < -a = |a| < |b|\) holds, which implies \(b \ne 0\). Now, we show that both \(b < 0\) and \(b > 0\) lead to a contradiction. If \(b < 0\), then \(|b| = -b\), and thus it holds \(-a < -b\), which amounts to \(a > b\), contradicting the fact that [a, b] is an interval. If \(b > 0\), then \(|b| = b\), and thus \(-a < b\) holds, which contradicts the hypothesis that \(-a > b\). This proves the two claims above. Now, we distinguish the following sub-cases.

If \(X = O\), then [c, d] is such that \(a < c < b < d\). We distinguish the following cases.

If \(-c > d\), then take some \(c'\) such that \(a' < c' < -|b'| < 0\) (notice also that \(c' < -|b'| \le b'\) trivially holds), and \(d'\) such that \(b' < d' < |c'| = -c'\) (the existence of such points \(c',d'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_O [c',d']\) and \([c,d] Z [c',d']\).

If \(-c = d\), then take some \(c'\) such that \(a' < c' < -|b'| < 0\), and \(d' = -c'\) (the existence of such a point \(c'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_O [c',d']\) and \([c,d] Z [c',d']\).

If \(-c < d\), then take \(c'\) such that \(a' < c' < -|b'| < 0\), and any \(d' > -c'\) (the existence of such a point \(c'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_O [c',d']\) and \([c,d] Z [c',d']\).

If \(X = \overline{B}\), then [c, d] is such that \(a = c < b < d\). We distinguish the cases below.

If \(-c > d\), then take \(c' = a'\) and \(d'\) such that \(b' < d' < -a' = -c'\) (the existence of such a point \(d'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_{\overline{B}} [c',d']\) and \([c,d] Z [c',d']\).

If \(-c = d\), then take \(c' = a'\) and \(d' = -c' (= -a' > b')\). The interval \([c',d']\) is such that \([a',b'] R_{\overline{B}} [c',d']\) and \([c,d] Z [c',d']\).

If \(-c < d\), then take \(c' = a'\) and any \(d' > -c' (= -a' > b')\). The interval \([c',d']\) is such that \([a',b'] R_{\overline{B}} [c',d']\) and \([c,d] Z [c',d']\).

If \(X = \overline{O}\), then [c, d] is such that \(c < a < d < b\). We distinguish the cases below.

If \(-c < d\), then take some \(d'\) and \(c'\) such that \(|a'| < d' < |b'| = b'\) and \(-d' < c' < |a'| = -c\) (the existence of points \(c', d'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_{\overline{O}} [c',d']\) and \([c,d] Z [c',d']\).

If \(-c = d\), then take some \(d'\) such that \(|a'| < d' < |b'| = b'\) and \(c' = -d'\) (the existence of such a point \(d'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_{\overline{O}} [c',d']\) and \([c,d] Z [c',d']\).

If \(-c > d\), then take some \(d'\) and \(c'\) such that \(|a'| < d' < |b'| = b'\) and \(c' < -d'\) (the existence of points \(c', d'\) is guaranteed by the left-unboundedness and the density of \({\mathbb {R}}\), respectively). The interval \([c',d']\) is such that \([a',b'] R_{\overline{O}} [c',d']\) and \([c,d] Z [c',d']\).

If \(X = \overline{E}\), then [c, d] is such that \(c < a < b = d\). We distinguish the following cases.

If \(-c < d\), then take \(d' = b'\) and some \(c'\) such that \(-d' < c' < a'\) (the existence of such a point \(c'\) is guaranteed by the density of \({\mathbb {R}}\)). The interval \([c',d']\) is such that \([a',b'] R_{\overline{E}} [c',d']\) and \([c,d] Z [c',d']\).

If \(-c = d\), then take \(d' = b'\) and \(c' = -d' (= -b' < a')\). The interval \([c',d']\) is such that \([a',b'] R_{\overline{E}} [c',d']\) and \([c,d] Z [c',d']\).

If \(-c > d\), then take \(d' = b'\) and any \(c' < -d' (= -b' < a')\). The interval \([c',d']\) is such that \([a',b'] R_{\overline{E}} [c',d']\) and \([c,d] Z [c',d']\).

If \(X = O\), then [c, d] is such that \(a < c < b < d\). Notice that \(-d < -b < a < c\). Then, take some \(c'\) such that \(a' < c' < b'\) (the existence of such a point \(c'\) is guaranteed by the density of \({\mathbb {R}}\)) and any \(d' > b'\). It holds that \(c' > a' > -b' > -d'\). The interval \([c',d']\) is such that \([a',b'] R_{O} [c',d']\) and \([c,d] Z [c',d']\).

Since the relation Z is symmetric, by Proposition 2 we have that the backward condition is verified, too. Therefore, Z is an \(\mathsf {O \overline{B E O}}\)-bisimulation that violates \(\langle L \rangle \), and the thesis follows. \(\square \)

Moreover, let Z be a relation between (intervals of) \(M_1\) and \(M_2\) defined as follows: \([x,y]Z[w,z] {\Leftrightarrow } [x,y] \in V_1(p)\) if and only if \([w,z] \in V_2(p)\). It is easy to verify that [0, 3]Z[0, 3], \(M_1,[0,3] \Vdash \langle E \rangle p\), but \(M_2,[0,3] \Vdash \lnot \langle E \rangle p\). We show now that Z is an \(\mathsf {A B D O \overline{A B E}}\)-bisimulation between \(M_1\) and \(M_2\). The local condition immediately follows from the definition. As for the forward condition, it can be checked as follows. Let [x, y] and [w, z] be two Z-related intervals, and let us assume that \([x,y] R_X [x',y']\) holds for some \(X \in \{ A, B, D, O, \overline{A}, \overline{B}, \overline{E} \}\). We have to exhibit an interval \([w',z']\) such that \([x',y']\) and \([w',z']\) are Z-related, and [w, z] and \([w',z']\) are \(R_X\)-related. We proceed by considering each case in turn.

If \(X=A\), then \(y = x'\). We can always find a point \(z'\) such that \(z' > \max \{ 3, z \}\) and \(z' \in {\mathbb {Q}}\) if and only if \(y' \in {\mathbb {Q}}\) (since both \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\) are right-unbounded). This implies that \([x',y']\) and \([z,z']\) are Z-related. Since [w, z] and \([z,z']\) are obviously \(R_A\)-related, we have the thesis.

If \(X=B\), the argument is similar to the previous one, but, in this case, the density of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\) plays a major role. We choose a point \(z'\) such that \(w < z' < z\), \(z' \ne 3\), and \(z' \in {\mathbb {Q}}\) if and only if \(y' \in {\mathbb {Q}}\). The interval \([w,z']\) is such that \([x',y']\) and \([w,z']\) are Z-related, and [w, z] and \([w,z']\) are \(R_B\)-related.

If \(X = D\), it suffices to choose two points \(w'\) and \(z'\) such that \(w < w' < z' < z\), \(z' \ne 3\), \(w'\) belongs to \({\mathbb {Q}}\) if and only if \(x'\) does, and \(z'\) belongs to \({\mathbb {Q}}\) if and only if \(y'\) does. The existence of such points is guaranteed by the density of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\). The interval \([w',z']\) is such that \([w,z] R_D [w',z']\) and \([x',y'] Z [w',z']\).

If \(X=O\), then \(w'\) and \(z'\) are required to be such that \(w < w' < z < z'\), and both density and right-unboundedness of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\) must be exploited in order to choose a point \(w'\) such that \(w < w' < z\) and \(w' \in {\mathbb {Q}}\) if and only if \(x'\) does, and a point \(z'\) such that \(z' > \max \{ 3, z \}\) and \(z'\) belongs to \({\mathbb {Q}}\) if and only if \(y'\) does. The interval \([w',z']\) is such that \([w,z] R_O [w',z']\) and \([x',y'] Z [w',z']\).

If \(X=\overline{A}\), then there exists a point \(w''\) such that \(w'' < \min \{ 0, w \}\) and \(w'' \in {\mathbb {Q}}\) if and only if w does (and thus \(M', [w'',w] \Vdash p\)) and there exists a point \(w'''\) such that \(w''' < w\) and \(w''' \in {\mathbb {Q}}\) if and only if \(w \in \overline{{\mathbb {Q}}}\) (and thus \(M', [w''',w] \Vdash \lnot p\)). We choose \(w' = w''\) if \(M,[x',y'] \models p\), otherwise we choose \(w' = w'''\). The interval \([w',w]\) is such that \([w,z] R_{\overline{A}} [w',w]\) and \([x',y'] Z [w',w]\).

If \(X=\overline{B}\), then there exists a point \(z''\) such that \(z'' > \max \{ 3, z \}\) and \(z'' \in {\mathbb {Q}}\) if and only if w does (and thus \(M', [w,z''] \Vdash p\)) and there exists a point \(z'''\) such that \(z''' > z\) and \(z''' \in {\mathbb {Q}}\) if and only if \(w \in \overline{{\mathbb {Q}}}\) (and thus \(M', [w,z'''] \Vdash \lnot p\)). We choose \(z' = z''\) if \(M,[x',y'] \models p\), otherwise we choose \(z' = z'''\). The interval \([w,z']\) is such that \([w,z] R_{\overline{B}} [w,z']\) and \([x',y'] Z [w,z']\).

If \(X=\overline{E}\), then there exists a point \(w''\) such that \(w'' < \min \{ 0, w \}\) and \(w'' \in {\mathbb {Q}}\) if and only if z does (and thus \(M', [w'',z] \Vdash p\)) and there exists a point \(w'''\) such that \(w''' < w\) and \(w''' \in {\mathbb {Q}}\) if and only if \(z \in \overline{{\mathbb {Q}}}\) (and thus \(M', [w''',z] \Vdash \lnot p\)). We choose \(w' = w''\) if \(M,[x',y'] \models p\), otherwise we choose \(w' = w'''\). The interval \([w',z]\) is such that \([w,z] R_{\overline{E}} [w',z]\) and \([x',y'] Z [w',z]\).

Proof

Let \(M_1 = \langle {\mathbb {I}}({\mathbb {R}}), V_1 \rangle \) and \(M_2 = \langle {\mathbb {I}}({\mathbb {R}}), V_2 \rangle \) be two models built on the only proposition letter p. In order to define the valuation functions \(V_1\) and \(V_2\), we make use of two partitions of the set \({\mathbb {R}}\), one for \(M_1\) and the other for \(M_2\), each of them consisting of four sets that are dense in \({\mathbb {R}}\). Formally, for \(j = 1, 2\) and \(i = 1, \ldots , 4\), let \({\mathbb {R}}_j^i\) be dense in \({\mathbb {R}}\). Moreover, for \(j = 1, 2\), let \({\mathbb {R}} = \bigcup _{i=1}^4 {\mathbb {R}}_j^i\) and \({\mathbb {R}}_j^i \cap {\mathbb {R}}_j^{i'} = \emptyset \) for each \(i,i' \in \{ 1,2,3,4 \}\), with \(i \ne i'\). For the sake of simplicity, we impose the two partitions to be equal and thus we can safely omit the subscript, that is, \({\mathbb {R}}_1^{i} = {\mathbb {R}}_2^{i} = {\mathbb {R}}^{i}\) for each \(i \in \{ 1,2,3,4 \}\). Thanks to this condition, the bisimulation relation Z, that we define below, is symmetric. We force points in \({\mathbb {R}}^1\) (resp., \({\mathbb {R}}^2\), \({\mathbb {R}}^3\), \({\mathbb {R}}^4\)) to behave in the same way with respect to the truth of \(p / \lnot p\) over the intervals they initiate and terminate by imposing the following constraints. For \(j = 1,2\):

It is worth pointing out that two intervals [x, y] and [w, z] that are Z-related are such that if, for instance, both x and w belong to \({\mathbb {R}}^3\) (second clause), then either y and z both occur in odd-numbered partitions or they both occur in even-numbered partitions. Moreover, since the two partitions are equal, Z is symmetric.

Let us consider now two intervals [x, y] and [w, z] such that \(x \in {\mathbb {R}}^1\), \(w \in {\mathbb {R}}^1\), \(y \in {\mathbb {R}}^3\), and \(z \in {\mathbb {R}}^1\). By definition of Z, [x, y] and [w, z] are Z-related, and by definition of \(V_1\) and \(V_2\), there exists \(y'> y\) such that \(M_1,[y,y'] \Vdash p\), but there is no \(z' > z\) such that \(M_2,[z,z'] \Vdash p\). Thus, \(M_1,[x,y] \Vdash \langle A \rangle p\) and \(M_2,[w,z] \Vdash \lnot \langle A \rangle p\) hold.

To complete the proof, it suffices to show that the relation Z is a \(\mathsf {B E \overline{A B E}}\)-bisimulation. It can be easily checked that every pair ([x, y], [w, z]) of Z-related intervals is such that either \([x,y] \in V_1(p)\) and \([w,z] \in V_2(p)\), or \([x,y] \not \in V_1(p)\) and \([w,z] \not \in V_2(p)\).

In order to verify the forward condition, let [x, y] and [w, z] be two Z-related intervals. For each modality \(\langle X \rangle \) of the language and each interval \([x',y']\) such that \([x,y] R_X [x',y']\), we have to exhibit an interval \([w',z']\) such that \([x',y']Z[w',z']\) and \([w,z]R_X[w',z']\). We proceed by considering each case in turn.

The backward condition follows from the forward one by Proposition 2. Therefore, Z is a \(\mathsf {B E\overline{A B E}}\)-bisimulation that violates \(\langle A \rangle \), and the thesis immediately follows. \(\square \)

Let us consider the interval [0, 3] in \(M_{1}\) and the interval [0, 3] in \(M_{2}\). It is immediate to see that these two intervals are Z-related. However, \(M_{1},[0,3] \Vdash \langle D \rangle p\) (as \(M_{1},[1,2] \Vdash p\)), but \(M_{2},[0,3] \Vdash \lnot \langle D \rangle p\).

To complete the proof, it suffices to show that Z is an \(\mathsf {A B O\overline{A B E}}\)-bisimulation between \(M_{1}\) and \(M_{2}\).

Let [x, y] and [w, z] be two Z-related intervals. By definition, \(y \sim f_1(x)\) and \(z \sim f_2(w)\) for some \(\sim \in \{ <,=,\)\(> \}\). If \(\sim \in \{=,>\}\), then both [x, y] and [w, z] satisfy p; otherwise, both of them satisfy \(\lnot p\). Thus, the local condition is satisfied.

As for the forward condition, let [x, y] and \([x',y']\) be two intervals in \(M_{1}\) and [w, z] an interval in \(M_{2}\). We have to prove that if [x, y] and [w, z] are Z-related, then, for each modality \(\langle X \rangle \) of \(\mathsf {A B O \overline{A B E}}\) such that \([x,y] R_X [x',y']\), there exists an interval \([w',z']\) such that \([x',y']\) and \([w',z']\) are Z-related and \([w,z] R_X [w',z']\). Once again, we proceed by examining each case in turn.

Let \(X = A\). By definition of \(\langle A \rangle \), \(x' = y\) and we are forced to choose \(w' = z\). By \(y \equiv z\), it immediately follows that \(x' \equiv w'\). We must find a point \(z'>z\) such that \(y' \equiv z'\) and both \(y' \sim f_1(y)\) and \(z' \sim f_2(z)\) for some \(\sim \in \{<,=,>\}\). Let us suppose that \(y' < f_1(y)\). In such a case, we choose a point \(z'\) such that \(z < z' < f_2(z)\) and \(y' \equiv z'\). The existence of such a point is guaranteed by property (i) of \(f_2\) above and by the density of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\) in \({\mathbb {R}}\). Otherwise, if \(y' = f_1(y)\), we choose \(z' = f_2(z)\). By definition of \(f_1\) and \(f_2\) (the codomain of \(f_1\) and \(f_2\) is \({\mathbb {Q}}\)), both \(y'\) and \(z'\) belong to \({\mathbb {Q}}\) and thus \(y' \equiv z'\). Finally, if \(y' > f_1(y)\), we choose \(z' > f_2(z)\) such that \(y' \equiv z'\). The existence of such a point is guaranteed by right-unboundedness of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\), and the interval \([z,z']\) is such that \([x',y']Z[z,z']\) and \([w,z] R_A [z,z']\).

Let \(X = B\). In this case, \(x = x'\) and \(y' < y\). We distinguish the following cases.

If \(y' < f_1(x)\) and \(y' \in {\mathbb {Q}}\) (resp., \(y' \in \overline{{\mathbb {Q}}}\)), then we choose a point \(z' \in {\mathbb {Q}}\) (resp., \(z' \in \overline{{\mathbb {Q}}}\)) such that \(w < z' < \min \{ z, f_2(w) \}\) (the existence of such a point is guaranteed by density of \({\mathbb {Q}}\) and \(\overline{{\mathbb {Q}}}\), respectively).

In all cases, the interval \([w,z']\) is such that \([x,y']Z[w,z']\) and \([w,z] R_B [w,z']\).

Let \(X = O\). Firstly, we choose a point \(w'\) such that \(w < w' < z\), \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\), and \(f_2(w') > z\) (the existence of such a point is guaranteed by property (iii) of \(f_2\) on page 19). Secondly, we choose a point \(z'\) such that \(z' \in {\mathbb {Q}}\) iff \(y' \in {\mathbb {Q}}\), and

if \(y' < f_1(x')\), then \(z < z' < f_2(w')\) (density of \({\mathbb {Q}}\) and \(\overline{{\mathbb {Q}}}\) is used here),

if \(y' > f_1(x')\), then \(z' > f_2(w')\) (right-unboundedness of \({\mathbb {Q}}\) and \(\overline{{\mathbb {Q}}}\) is used here),

if \(y' = f_1(x')\), then \(z' = f_2(w')\).

In all cases, the interval \([w',z']\) is such that \([x',y']Z[w',z']\) and \([w,z] R_O [w',z']\).

Let \(X = \overline{A}\). In this case, \(y' = x\). We distinguish the following cases.

In all cases, the interval \([w',z]\) is such that \([x',y]Z[w',z]\) and \([w,z] R_{\overline{E}} [w',z]\).

The backward condition can be verified in a very similar way and thus the details of the proof are omitted. Hence, Z is an \(\mathsf {A B O\overline{A B E}}\)-bisimulation that violates \(\langle D \rangle \), hence the thesis. \(\square \)

We show that Z is an \(\mathsf {A B E \overline{A E D}}\)-bisimulation between \(M_1\) and \(M_2\). The local condition immediately follows from the definition. As for the forward condition, it can be checked as follows. Let [x, y] and [w, z] be two Z-related intervals, and let us assume that \([x,y] R_X [x',y']\) holds for some \(X \in \{ A, B, E, \overline{A}, \overline{E}, \overline{D} \}\). We have to exhibit an interval \([w',z']\) such that \([x',y']\) and \([w',z']\) are Z-related, and [w, z] and \([w',z']\) are \(R_X\)-related. We proceed by a case analysis on \(X \in \{ A, B, E, \overline{A}, \overline{E}, \overline{D} \}\).

If \(X=A\), then we distinguish the following cases: (a) if \(0 < z < 3\), then we select a point \(z'\) such that \(z < z' < 3\) and \(z' \in {\mathbb {Q}}\) iff \(y' \in {\mathbb {Q}}\) (the existence of such a point is guaranteed by density of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\)); (b) otherwise, we select a point \(z'\) such that \(z' > z\) and \(z' \in {\mathbb {Q}}\) iff \(y' \in {\mathbb {Q}}\) (the existence of such a point is guaranteed by right-unboundedness of \({\mathbb {Q}}\) and \(\overline{\mathbb {Q}}\)). In both cases, the interval \([z,z']\) is such that \([x',y']Z[z,z']\) and \([w,z]R_A[z,z']\).

If \(X=B\), the argument is similar to the previous one. We distinguish the following cases: (a) if \(0 < w < 3\), then we choose a point \(z'\) such that \(w < z' < \min \{ 3,z \}\) and \(z' \in {\mathbb {Q}}\) iff \(y' \in {\mathbb {Q}}\); (b) otherwise, we choose a point \(z'\) such that \(w < z' < z\) and \(z' \in {\mathbb {Q}}\) iff \(y' \in {\mathbb {Q}}\). In both cases, the interval \([w,z']\) is such that \([x',y']Z[w,z']\) and \([w,z]R_B[w,z']\).

If \(X = E\), then we distinguish the following cases: (a) if \(z > 3\), then we choose a point \(w'\) such that \(\max \{ 3,w \} < w' < z\) and \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\); (b) otherwise, we choose a point \(w'\) such that \(w < w' < z\) and \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\). In both cases, the interval \([w',z]\) is such that \([x',y']Z[w',z]\) and \([w,z]R_E[w',z]\).

If \(X = \overline{A}\), then we choose a point \(w'\) such that \(w' < \min \{ 0,w \}\) and \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\). The interval \([w',w]\) is such that \([x',y']Z[w',w]\) and \([w,z]R_{\overline{A}}[w',w]\).

If \(X = \overline{E}\), then we choose a point \(w'\) such that \(w' < \min \{ 0,w \}\) and \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\). The interval \([w',z]\) is such that \([x',y']Z[w',z]\) and \([w,z]R_{\overline{E}}[w',z]\).

If \(X = \overline{D}\), then we first choose a point \(w'\) such that \(w' < \min \{ 0,w \}\) and \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\). Next, we choose a point \(z'\) such that \(z' > z\) and \(z' \in {\mathbb {Q}}\) iff \(y' \in {\mathbb {Q}}\). The interval \([w',z']\) is such that \([x',y']Z[w',z']\) and \([w,z]R_{\overline{D}}[w',z']\).

The backward condition can be verified in a very similar way and thus we omit the details of the proof. Therefore, Z is an \(\mathsf {A B E\overline{A E D}}\)-bisimulation that violates \(\langle O \rangle \). The thesis immediately follows. \(\square \)

Finally, for each \(i \in \{1,2\}\), we use \(\overline{\mathcal {S}}_i\) to denote the set \({\mathbb {I}}({\mathbb {R}}) \setminus {\mathcal {S}}_i\). It is easy to verify that, for every pair of points \(x,y \in {\mathbb {I}}({\mathbb {R}})\), if \(x < y\), then there exist \(y_1,y_2,y_3,y_4 \in {\mathbb {R}}\) such that \(x < y_i < y\), for each \(i \in \{1,2,3,4\}\), and:

We define now a pair of functions that will be used in the definition of the models involved in the bisimulation relation Z. Let \(g: {\mathbb {R}} \rightarrow {\mathbb {Q}}\) be a function defined as follows (notice the strong similarity with the definition of g in Lemma 12): for each \(x \in {\mathbb {R}}\), \(g(x) = q\), where \(q \in {\mathbb {Q}}\) is the unique rational number such that \(x \in {\mathbb {R}}_{q}^a \cup R_{q}^b\). The functions \(f_1: {\mathbb {R}} \rightarrow {\mathbb {Q}}\) and \(f_2: {\mathbb {R}} \rightarrow {\mathbb {Q}}\) are defined as follows:

where \(a_{n'}\) is the least element of the series \(a_n = 3-(\frac{1}{n})\) (\(n\ge 1\)) such that \(x < a_{n'}\). It is not hard to verify that the functions \(f_i\) (\(i \in \{ 1,2 \}\)) fulfill the following conditions:

At this point, we are ready to define the models \(M_1\) and \(M_2\), and the bisimulation relation between their intervals. Let \(i \in \{1,2\}\) and \(M_{i} = \langle {\mathbb {I}}({\mathbb {R}}), V_{f_i} \rangle \), where the valuation functions \(V_{i} : {{\mathcal {AP}}} \rightarrow 2^{{\mathbb {I}}({\mathbb {R}})}\) is defined as follows:

Now, by the definition of Z, we have that [0, 3]Z[0, 3] (notice that this is also a consequence of the facts that \(f_1(0) = f_2(0)\) and that \([0,3]R_O,[0,3]\) does not hold). Moreover, it is easy to see that \(M_{1},[0,3] \Vdash \langle O \rangle p\), while \(M_{2},[0,3] \Vdash \lnot \langle O \rangle p\) (this is a direct consequence of property (iv) of \(f_2\) and of the fact that \(f_1(x) > 3\) for some \(x \in (0,3)\)).

We show that Z is an \(\mathsf {A B D\overline{A B E}}\)- bisimulation. For the local condition, consider two intervals [x, y] and [w, z] such that [x, y]Z[w, z]. First, we assume that \([x,y] \in V_1(p)\) and we show that \([w,z] \in V_2(p)\) follows. Since \([x,y] \in V_1(p)\), either \(y = f_1(x)\) holds or both \(y < f_1(x)\) and \([x,y] \in {\mathcal {S}}_1\) hold. In the former case, by the definition of Z, it must be \(z = f_2(w)\), which implies \([w,z] \in V_2(p)\). In the latter case, by the definition of Z, both \(z < f_2(w)\) and \([w,z] \in {\mathcal {S}}_2\) hold, and thus \([w,z] \in V_2(p)\). Second, we assume that \([w,z] \in V_2(p)\) and we show that \([x,y] \in V_1(p)\) follows. Since \([w,z] \in V_2(p)\), either \(z = f_2(w)\) holds or both \(z < f_2(w)\) and \([w,z] \in {\mathcal {S}}_2\) hold. In the former case, by the definition of Z, it must be \(y = f_1(x)\), which implies \([x,y] \in V_1(p)\). In the latter case, by the definition of Z, both \(y < f_1(x)\) and \([x,y] \in {\mathcal {S}}_1\) hold, and thus \([x,y] \in V_1(p)\).

In order to prove that the forward condition is satisfied, we assume that [x, y]Z[w, z] and \([x,y] R_X [x',y']\), for some \(X \in \{ A, B, D, \overline{A}, \overline{B}, \overline{E} \}\) and some \([x,y], [w,z], [x',y'] \in {\mathbb {I}}({\mathbb {R}})\), and we show the existence of an interval \([w',z']\) such that \([x',y']Z[w',z']\) and \([w,z] R_X [w',z']\). As usual, we proceed by considering each case in turn.

In all cases, the interval \([w,z']\) is such that \([x',y']Z[w,z']\) and \([w,z] R_B [w,z']\).

If \(X = D\), then we first select a point \(w'\) such that \(w < w' < z\), \(w' \in {\mathbb {Q}}\) iff \(x' \in {\mathbb {Q}}\), and \(f_2(w') < z\) (the existence of such a point is guaranteed by property (iii) of \(f_2\)). Then, we select a point \(z'\) as follows.

In all cases, the interval \([w',z']\) is such that \([x',y']Z[w',z']\) and \([w,z] R_D [w',z']\).

If \(X = \overline{A}\), then we distinguish three cases.

If \(y' > f_1(x')\) and \(x' \in {\mathbb {Q}}\) (resp., \(x' \in \overline{{\mathbb {Q}}}\)), then consider a point \(\overline{z} \in {\mathbb {Q}}\) such that \(\overline{z} < w\). We select a point \(w' \in {\mathbb {Q}}\) (resp., \(w' \in \overline{{\mathbb {Q}}}\)) such that \(w' < \overline{z} < w\) and \(f_2(w') = \overline{z}\) (the existence of such a point is guaranteed by property (ii) of \(f_2\)).

In all cases, the interval \([w,z']\) is such that \([x',y']Z[w,z']\) and \([w,z] R_{\overline{B}} [w,z']\).

If \(X = \overline{E}\), then we distinguish three cases.

If \(y' > f_1(x')\) and \(x' \in {\mathbb {Q}}\) (resp., \(x' \in \overline{{\mathbb {Q}}}\)), then consider a point \(\overline{z} \in {\mathbb {Q}}\) such that \(\overline{z} < z\). We select a point \(w' \in {\mathbb {Q}}\) (resp., \(w' \in \overline{{\mathbb {Q}}}\)) such that \(w' < w\) and \(f_2(w') = \overline{z}\) (the existence of such a point is guaranteed by property (ii) of \(f_2\)).

If \(y' < f_1(x')\) and \(x' \in {\mathbb {Q}}\) (resp., \(x' \in \overline{{\mathbb {Q}}}\)), then consider a point \(\overline{z} \in {\mathbb {Q}}\) such that \(\overline{z} > z\). We select a point \(w' \in {\mathbb {Q}}\) (resp., \(w' \in \overline{{\mathbb {Q}}}\)) such that \(w' < \min \{ 0,w \}\), \(f_2(w') = \overline{z}\), and that \([w',w] \in {\mathcal {S}}_2\) if and only if \([x',y'] \in {\mathcal {S}}_1\) (the existence of such a point is guaranteed by property (ii) of \(f_2\)). Notice that, since \(w' < 0\), it is not the case that \([0,3] R_O [w',z]\).

In all cases, the interval \([w',z]\) is such that \([x',y']Z[w',z]\) and \([w,z] R_{\overline{E}} [w',z]\).

The backward condition can be verified in a very similar way and thus we omit the details of the proof. Therefore, Z is an \(\mathsf {A B D\overline{A B E}}\)-bisimulation that violates \(\langle O \rangle \), hence the thesis. \(\square \)